Version of February 23, 2005
Some New Results in Multiplicative
and Additive Ramsey Theory
Mathias Beiglböck1
Vitaly Bergelson2
Neil Hindman2
and
Dona Strauss
Abstract. There are several notions of largeness that make sense in any semigroup,
and others such as the various kinds of density that make sense in sufficiently well
behaved semigroups including (N, +) and (N, ·). It is known that sets with positive
multiplicative density must contain arbitrarily large geoarithmetic progressions, that
is, sets of the form {r j (a + id) : i, j ∈ {0, 1, . . . , k}}. We establish some combined
additive and multiplicative Ramsey Theoretic consequences of known algebraic results
in the semigroups (βN, +) and (βN, ·), derive some new algebraic results, and derive
consequences of them involving geoarithmetic progressions. For example, we show that
in any finite partition of N there must be, for each k, sets of the form {b(a + id) j : i, j ∈
{0, 1, . . . , k}} together with d, the arithmetic progression {a + id : i ∈ {0, 1, . . . , k}},
and the geometric progression {bdj : j ∈ {0, 1, . . . , k}} in one cell of the partition.
1. Introduction
Our starting point is the famous theorem of van der Waerden [19] which says that
whenever the set N of positive integers is divided into finitely many classes, one of these
classes contains arbitrarily long arithmetic progressions. The corresponding statement
about geometric progressions is easily seen to be equivalent via the homomorphisms
b : (N, +) → (N, ·) and ` : (N \ {1}, ·) → (N, +) where by b(n) = 2n and `(n) is the
length of the prime factorization of n.
In 1975, Szemerédi [18], showed that any set with positive upper asymptotic density contains arbitrarily long arithmetic progressions. (An ergodic theoretic proof of
Szemerédi’s Theorem can be found in [5], [6] or [7].) It has recently been shown [1,
Theorem 1.3] that any set having positive multiplicative upper Banach density – the
1 This
author thanks the Austrian Science Foundation FWF for its support through Project no.
S8312. He also thanks Ohio State University for its hospitality in the spring of 2004 while much of this
research was being conducted.
2 These authors acknowledge support received from the National Science Foundation (USA) via
grants DMS-0345350 and DMS-0243586 respectively.
1
notion d∗m of Definition 2.1 below – must contain substantial combined additive and
multiplicative structure; in particular it must contain arbitrarily large geoarithmetic
progressions, that is, sets of the form r j (a + id) : i, j ∈ {0, 1, . . . , k} . (The results
in [1] actually require a property weaker than having d∗m positive.) In Corollary 3.8 we
show that one cell of any finite partition of N must satisfy a stronger conclusion than
this.
Another simply stated result from [1] is that any multiplicatively large set contains
geometric progressions in which the common ratios form an arithmetic progression, that
is a set of the form b(a + id)j : i, j ∈ {0, 1, . . . , k} . As a consequence, one cell of any
finite partition of N must satisfy this property. We provide in Corollary 4.6 a new
reasonably simple proof of this consequence.
In Sections 3 and 4 we shall be concerned with the question of what sort of combined
additive and multiplicative structures can be guaranteed to lie in one cell of a finite
partition of N. For example, it was shown in 1975 [10] that there exist sequences hx n i∞
n=1
∞
∞
and hyn i∞
n=1 such that F S(hxn in=1 ) ∪ F P (hyn in=1 ) is contained in one cell, where
P
Q
∞
F S(hxn i∞
n=1 ) = {
n∈F xn : F ∈ Pf (N)}, F P (hyn in=1 ) = { n∈F yn : F ∈ Pf (N)},
and for any set X, Pf (X) is the set of finite nonempty subsets of X.
In Section 3 we present some combined additive and multiplicative results that are
easily obtainable from known algebraic results, but do not seem to have been previously
stated. For example, the following is a special case of Corollary 3.7.
Sm
1.1 Theorem. Let m, k ∈ N and let N = i=1 Ai . Then there exist i ∈ {1, 2, . . . , m},
a, d, b ∈ Ai , and r ∈ Ai \ {1} such that
s
br
:
s
∈
{0,
1,
.
.
.
,
k}
∪
a
+
td
:
t
∈
{0,
1,
.
.
.
,
k}
∪ {rd}
s
∪
r(a + td): t ∈ {0, 1, . . . , k} ∪ bdr : s ∈ {0, 1, . . . , k} ∪
br s (a + td) : s, t ∈ {0, 1, . . . , k} ⊆ Ai .
In Section 4 we derive several new algebraic results and new combinatorial consequences thereof.
In particular, we have the following consequence of Corollary 4.4.
1.2 Theorem. Let m, k ∈ N and let N =
Sm
i=1
Ai . Then there exist i ∈ {1, 2, . . . , m}
and a, d, b ∈ Ai such that
b(a + id)j : i,j ∈ {0, 1, . . . , k} ∪ bdj: j ∈ {0, 1, . . . , k}
∪ a + di : i ∈ {0, 1, . . . , k} ⊆ Ai .
Consider now the following result, which is a consequence of [1, Theorem 3.13].
2
∞
1.3 Theorem. Let m, k ∈ N. For each i ∈ {0, 1, . . . , k} let hxi,t i∞
t=1 and hyi,t it=1 be
Sm
sequences in N. Let N = s=1 As . Then there exist s ∈ {1, 2, . . . , m}, F, G ∈ Pf (N),
P
Q
and a, b ∈ N such that b(a + t∈F xi,t ) · ( t∈G yj,t ) : i, j ∈ {0, 1, . . . , k} ⊆ As .
Notice that a particular consequence of Theorem 1.3 is that one cell of each finite
partition of N must contain arbitrarily long geoarithmetic progressions. Further, the
∞
common ratio r can be taken from F P (hyn i∞
n=1 ) for any prescribed hyn in=1 and the
additive increment d can be guaranteed to be a multiple of some member of F S(hxn i∞
n=1 )
for any prescribed hxn i∞
n=1 . To see this, for i ∈ {1, 2, . . . , k} and t ∈ N, let xi,t = ixt
P
i
and yi,t = (yt ) . Given F and G as guaranteed by Theorem 1.3, let d = b · t∈F xt and
Q
r = t∈G yt .
We show in Theorem 4.8 that one may take F = G in Theorem 1.3 and in Corollary
4.12 that one may eliminate b (and in particular, that the additive increment for the
geoarithmetic progressions described above can be taken from F S(hxn i∞
n=1 ) for any
hxn i∞
n=1 ). We show also that one may not simultaneously take F = G and eliminate
b. The example of Theorem 4.17 shows also that one cannot eliminate the multiplier b
and expect to find configurations if the sort given in Theorem 1.3 in all sets with d∗m
equal to 1.
The following consequence of Corollary 3.8 (or alternatively of Corollary 4.12(c) or
Corollary 4.23) says that one can always get the additive increment of a geoarithmetic
progression as the initial term of a geometric progression in one cell of a finite partition
of N.
Sm
1.4 Theorem. Let m ∈ N and let N = s=1 As . Then there exist s ∈ {1, 2, . . . , m},
a, d ∈ As and r ∈ As \ {1} such that r j (a + id) : i, j ∈ {0, 1, . . . , k} ∪ dr j : j ∈ {0, 1,
. . . , k} ⊆ As .
In Section 5 we establish some limitations on the algebraic approach and prove
a theorem which, for countable commutative semigroups, is even stronger than the
powerful Central Sets Theorem. (The Central Sets Theorem for the semigroup (N, +)
is [6, Proposition 8.21].) Central subsets of any semigroup are guaranteed substantial
combinatorial structure. See [13, Part III] for numerous examples. Several earlier results
in the paper follow immediately from this theorem. However, we prove the earlier results
directly instead of stating them as corollaries, because the direct proofs are reasonably
simple, while the theorem proved in Section 5 might be considered a little daunting.
3
2. Preliminaries
We shall be concerned with several notions of largeness, both additive and multiplicative.
Among these are various notions of density. The notions d and dm defined below are
referred to as upper asymptotic density and the notions d∗ and d∗m are called upper
Banach density.
2.1 Definition. Let A ⊆ N, and let hpn i∞
n=1 be the sequence of primes in their natural
order.
|A ∩ {1, 2, . . . , n}|
.
(a) d(A) = lim sup
n
n→∞
|A ∩ {m + 1, m + 2, . . . , n}|
(b)
d∗ (A) = lim sup
n−m
n−m→∞
|A ∩ (m + {1, 2, . . . , n})|
= lim sup
: m ∈ N and n ≥ k .
k→∞
n
Qn
αi
(c) For n ∈ N, Fn =
: for each i ∈ {1, 2, . . . , n}, αi ∈ {0, 1, . . . , n} .
i=1 pi
|A ∩ Fn |
(d) dm (A) = lim sup
.
|Fn |
n→∞
|A ∩ (m · Fn )|
∗
(e) dm (A) = lim sup
: m ∈ N and n ≥ k .
k→∞
|Fn |
The sequence hFn i∞
n=1 defined above is a Følner sequence in (N, ·). The notions d m
and d∗m could be defined in terms of any Følner sequence (with the values depending
on the choice of the Følner sequence). See [1, Section 2].
Other notions of largeness with which we shall be concerned originated in the
study of topological dynamics and make sense in any semigroup. Four of these, namely
thick , syndetic, piecewise syndetic and IP-set have simple elementary descriptions and
we introduce them now. The fifth, central is most simply described in terms of the
algebraic structure of βS, which we shall describe shortly. Given a semigroup (S, ·), a
subset A of S, and x ∈ S, we let x−1 A = {y ∈ S : xy ∈ A}.
2.2 Definition. Let (S, ·) be a semigroup and let A ⊆ S.
(a) A is thick if and only if whenever F ∈ Pf (S) there exists x ∈ S such that F x ⊆ A.
S
(b) A is syndetic if and only if there exists G ∈ Pf (S) such that S = t∈G t−1 A.
(c) A is piecewise syndetic if and only if there exists G ∈ Pf (S) such that for every
S
F ∈ Pf (S) there exists x ∈ S such that F x ⊆ t∈G t−1 A.
(d) A is an IP-set if and only if there exists a sequence hxn i∞
n=1 in S such that
F P (hxn i∞
n=1 ) ⊆ A.
4
Notice that each of thick and syndetic imply piecewise syndetic and thick sets are
IP-sets. It is easy to construct examples in (N, +) showing that no other implications
among these notions is valid in general.
The following Lemma gives a hint why piecewise syndetic sets will be interesting
for our purposes. A family A of subsets of a set X is partition regular provided that
whenever X is partitioned into finitely many classes, one of these classes contains a
member of A.
2.3 Lemma. Let (S, ·) be a semigroup, let F be a partition regular family of finite
subsets of S, and let A be a piecewise syndetic subset of S. Then there exist t, x ∈ S
and F ∈ F such that tF x ⊆ A. If (S, ·) is commutative, then there exist t ∈ S and
F ∈ F such that tF ⊆ A.
Proof. Pick G ∈ Pf (S) such that for every F ∈ Pf (S) there exists x ∈ S such that
S
S
F x ⊆ t∈G t−1 A. For each F ∈ Pf (S) choose xF ∈ S such that F xF ⊆ t∈G t−1 A
and linearly order G. Let ∞ be a point not in G and let K = G ∪ {∞} have the discrete
topology. For each F ∈ Pf (S) define ϕF ∈ ×s∈S K by ϕF (s) = min{t ∈ G : tsxF ∈ A}
if s ∈ F and ϕF (s) = ∞ if s ∈
/ F . Direct Pf (S) by inclusion and let ψ be a cluster
point of the net hϕF iF ∈Pf (S) in ×s∈S K.
S
Then S ⊆ t∈K ψ −1 [{t}]. Pick t ∈ K and F ∈ F such that F ⊆ ψ −1 [{t}]. Let
U = {τ ∈ ×s∈S K : for all s ∈ F , τ (s) = ψ(s)}. Then U is a neighborhood of ψ so
pick H ∈ Pf (S) such that F ⊆ H and ϕH ∈ U . Then for all s ∈ F , ϕH (s) = t so t ∈ G
and tF xH ⊆ A.
Notice that if (S, ·) is not commutative, then both multipliers in Lemma 2.3 may
be required. For example, let S be the free semigroup on the letters a and b. Then
F = {bF : F ∈ Pf (S)} and G = {F b : F ∈ Pf (S)} are partition regular, aS and Sa are
piecewise syndetic, there do not exist F ∈ F and x ∈ S with F x ⊆ aS, and there do
not exist F ∈ G and t ∈ S with tF ⊆ Sa. (In fact, aS is syndetic in S.)
In Section 3 we shall need to deal with the columns condition.
2.4 Definition. Let u, v ∈ N, let C be a u × v matrix with entries from Q, and let
c~1 , c~2 , . . . , c~v be the columns of C. Let R = Z or R = Q. The matrix C satisfies the
columns condition over R if and only if there exist m ∈ N and I1 , I2 , . . . , Im such that
(1) {I1 , I2 , . . . , Im } is a partition of {1, 2, . . . , v}.
P
~i = ~0.
(2)
i∈I1 c
5
(3) If m > 1 and t ∈ {2, 3, . . . , m}, let Jt =
P
P
such that i∈It c~i = i∈Jt δt,i · c~i .
St−1
j=1
Ij . Then there exist hδt,i ii∈Jt in R
In [17], Rado proved that a u × v matrix C is kernel partition regular over (N, +)
Sr
(meaning that whenever r ∈ N and N = i=1 Ai , there exist i ∈ {1, 2, . . . , r} and
~x ∈ Ai v such that C~x = ~0) if and only if C satisfies the columns condition over Q.
A u × v matrix C with entries from Q is image partition regular over (N, +) if and
Sr
only if whenever r ∈ N and N = i=1 Ai , there exist i ∈ {1, 2, . . . , r} and ~x ∈ Nv such
that all entries of C~x are in Ai . We shall use the custom of denoting the entries of a
matrix by the lower case of the same letter whose upper case denotes the matrix, so
that the entry in row i and column j of C is denoted by ci,j .
2.5 Definition. Let u, v ∈ N and let C be a u × v matrix with entries from Q.
(a) C is a first entries matrix if and only if now row of C is ~0 and for all i, j ∈ {1, 2,
. . . , u} and all k ∈ {1, 2, . . . , v}, if k = min{t : ci,t 6= 0} = min{t : cj,t 6= 0}, then
ci,k = cj,k > 0.
(b) The number b is a first entry of C if and only if b is the first nonzero entry in some
row of C.
Each first entries matrix is image partition regular over (N, +) and image partition regular matrices can be characterized in terms of first entries matrices. (See [13,
Theorem 15.24].)
We now present a brief review of basic facts about (βS, ·). For additional information and any unfamiliar terminology encountered see [13].
Given a discrete semigroup (S, ·) we take the points of the Stone-Čech compactification βS of S to be the ultrafilters on S, the principal ultrafilters being identified with
the points of S. Given A ⊆ S, A = {p ∈ βS : A ∈ p} and the set {A : A ⊆ S} is a basis
for the open sets (and a basis for the closed sets) of βS. Given p, q ∈ βS and A ⊆ S,
A ∈ p · q if and only if {x ∈ S : x−1 A ∈ q} ∈ p.
With this operation, (βS, ·) is a compact Hausdorff right topological semigroup
with S contained in its topological center. That is, for each p ∈ βS, the function
ρp : βS → βS defined by ρp (q) = q · p is continuous and for each x ∈ S, the function
λx : βS → βS defined by λx (q) = x · q is continuous. A subset I of a semigroup T is a
left ideal provided T · I ⊆ I, a right ideal provided I · T ⊆ I, and a two sided ideal (or
simply an ideal ) provided it is both a left ideal and a right ideal.
Any compact Hausdorff right topological semigroup T has a smallest two sided
S
S
ideal K(T ) = {L : L is a minimal left ideal of T } = {R : R is a minimal right ideal
6
of T }. Given a minimal left ideal L and a minimal right ideal R, L ∩ R is a group, and in
particular contains an idempotent. An idempotent in K(T ) is a minimal idempotent. If
p and q are idempotents in T we write p ≤ q if and only if pq = qp = p. An idempotent
is minimal with respect to this relation if and only if it is member of the smallest ideal.
A subset of S is an IP-set if and only if it is a member of some idempotent in βS.
2.6 Definition. Let S be a semigroup and let A ⊆ S. Then A is central if and only if
there is a minimal idempotent p of βS such that A ∈ p.
A central set is in particular a piecewise syndetic IP-set. Given a minimal idempotent p and a finite partition of S, one cell must be a member of p, hence at least one
cell of any finite partition of S must be central. Central sets are fundamental to the
Ramsey Theoretic applications of the algebra of βS.
We shall need the Hales-Jewett Theorem. Given the free semigroup S over an
alphabet L, a variable word w is a word over L ∪ {v} in which v occurs, where v is a
“variable” not in L. Given a variable word w and a ∈ L, w(a) is the word in S obtained
by replacing each occurrence of v by a.
2.7 Theorem (Hales-Jewett). Let L be a finite alphabet, let S be the free semigroup
Sm
over L, let m ∈ N, and let S = i=1 Ai . Then there exist i ∈ {1, 2, . . . , m} and a
variable word w such that {w(a) : a ∈ L} ⊆ Ai .
Proof. [9, Theorem 1], or see [8, Theorem 2.3] or [13, Theorem 14.7].
Applications which we will use later are the following theorems. These results are
well known among afficianados.
2.8 Theorem. Let (S, ·) be a commutative semigroup, let A be a piecewise syndetic
subset of S, let k ∈ N, and for i ∈ {1, 2, . . . , k} let hyi,n i∞
i=1 be a sequence in S. There
Q
exist F ∈ Pf (N) and b ∈ S such that {b} ∪ b t∈F yi,t : i ∈ {1, . . . , k} ⊆ A.
Proof. By virtue of Lemma 2.3 it is sufficient to show that the family
Q
{b} ∪ b · t∈F yi,t : i ∈ {1, . . . , k} : b ∈ S, F ∈ Pf (N)
is partition regular.
Let L = {0, 1, . . . , k} and let T be the free semigroup on the alphabet L. Let b0 ∈ S
be an arbitrary, fixed element. Given a word w = l1 l2 · · · ln of length n in S, define
Q
f (w) = b0 t∈{1,2,...,n},lt 6=0 ylt ,t if there exists some t ∈ {1, 2, . . . , n} such that lt 6= 0
and f (w) = b0 otherwise.
7
Sm
Consider a partition {A1 , A2 , . . . , Am } of S. Then T = s=1 f −1 [As ] so pick s ∈
{1, 2, . . . , m} and a variable word w = l1 l2 · · · ln (with each lt ∈ L ∪ {v}) such that
{w(i) : i ∈ L} ⊆ f −1 [As ].
Let F = {t ∈ {1, 2, . . . , n} : lt = v}, let G = {1, 2, . . . , n} \ F and let b = f w(0) .
Q
Q
Then b t∈F yi,t = f w(i) for i ∈ {1, 2, . . . , k} and thus {b} ∪ b t∈F yi,t : i ∈
{1, . . . , k} ⊆ As .
2.9 Corollary. Let (S, ·) be a commutative semigroup, let A be a piecewise syndetic
subset of S, let B be an IP-set in S, and let k ∈ N. There exist b ∈ S and r ∈ B such
that {b, br, br 2, . . . , br k } ⊆ A. If A is central we may in particular take A = B, such
that {r, b, br, br 2, . . . , br k } ⊆ A.
∞
Proof. Let hxn i∞
n=1 be a sequence in S such that F P (hxn in=1 ) ⊆ B. For i ∈ {1, 2, . . . ,
k} and n ∈ N, let yi,n = (xn )i . Pick b and F as guaranteed by Theorem 2.8 and let
Q
r = t∈F xt .
Any central set is a piecewise syndetic IP-set and thus the in particular statement
follows.
3. New wine from old wineskins
All of the results about the algebraic structure of βN that are used in this section have
been known for several years.
There is a long list of configurations which are known to be present in any central
subset of (N, +) and a somewhat shorter, but still lengthy, list of structures which can
be found in any central subset of (N, ·). Some of these involve special subsets of βN
defined by various notions of density.
3.1 Definition.
(a) ∆ = {q ∈ βN : (∀A ∈ q)(d(A) > 0)}.
(b) ∆∗ = {q ∈ βN : (∀A ∈ q)(d∗ (A) > 0)}.
(c) ∆m = {q ∈ βN : (∀A ∈ q)(dm (A) > 0)}.
(d) ∆∗m = {q ∈ βN : (∀A ∈ q)(d∗m (A) > 0)}.
We summarize some of the structures guaranteed to be present in any multiplicatively central set first. See [13, Chapter 14] for a formalization of the notion of tree in
a set as well as the set of successors to a node.
8
3.2 Theorem. Let A be a central subset of (N, ·).
(a) For any sequence hxn i∞
n=1 in N and any k ∈ N, there exist b ∈ N and r ∈
∞
F P (hxn in=1 ) such that {b, br, br 2, . . . , br k } ⊆ A.
(b) There is a tree T in A such that for any path g through T , F P (hg(n)i ∞
n=1 ) ⊆ A and
for every node f ∈ T , the set Bf of successors to f satisfies d∗m (Bf ) > 0.
(c) If u, v ∈ N and C is a u × v matrix with entries from Z which satisfies the columns
condition over Z, then there exists ~x ∈ Av such that for all i ∈ {1, 2, . . . , u},
Qv
ci,j
= 1.
j=1 xj
(d) If u, v ∈ N and C is a u × v first entries matrix with entries from Z and all first
entries equal to 1, then there exists ~x in Nv such that for all i ∈ {1, 2, . . . , u},
Qv
ci,j
∈ A.
j=1 xj
Proof. (a) Corollary 2.9.
(b) Pick a minimal idempotent q of (βN, ·) such that A ∈ q. By [13, Theorems 20.5
and 20.6] ∆∗m is an ideal of (βN, ·), so q ∈ ∆∗m and [13, Lemma 14.24] applies.
(c) [13, Theorem 15.16(a)].
(d) [13, Lemma 15.14 and Theorem 15.5].
The conditions of Theorem 3.2(c) and (d) are stronger than those required for
kernel and image partition regularity over (N, ·). (And necessarily so. The set A =
N \ {x2 : x ∈ N} is central in (N, ·) [13, Exercise 15.1.2], the matrix ( 2 −2 1 ) is
kernel partition regular over (N, ·), and the matrix (2) is image partition regular over
(N, ·). But one cannot get x, y, z ∈ A with x2 y −2 z = 1 and one cannot get x ∈ N
with x2 ∈ A.) By contrast, in (N, +), kernel partition regularity of C corresponds to
solutions to C~x = ~0 in any central set and image partition regularity of C corresponds
to obtaining all entries of C~x in any central set.
We shall be interested in a property stronger than central for our additive results.
By [13, Theorem 6.79], ∆ is a compact left ideal of (βN, +) so contains a minimal idempotent of (βN, +). Consequently, any finite partition of N will have one cell satisfying
the hypothesis of the following theorem.
3.3 Theorem. Let A ⊆ N and assume that there is a minimal idempotent q of (βN, +)
in A ∩ ∆.
(a) For any sequence hxn i∞
n=1 in N and any k ∈ N, there exist a ∈ N and d ∈
F S(hxn i∞
n=1 ) such that {a, a + d, . . . , a + kd} ⊆ A.
9
(b) There is a tree T in A such that for any path g through T , F S(hg(n)i ∞
n=1 ) ⊆ A and
for every node f ∈ T , the set Bf of successors to f satisfies d(Bf ) > 0.
(c) If u, v ∈ N and C is a u × v matrix with entries from Q which is kernel partition
regular over (N, +) (that is C satisfies the columns condition over Q), then there
exists ~x ∈ Av such that C~x = ~0.
(d) If u, v ∈ N and C is a u × v matrix with entries from Q which is image partition
regular over (N, +), (in particular if C is a first entries matrix), then there exists
~x in Nv with all entries of C~x in A.
(e) Let R be a finite set of polynomials which take integer values at integers and have
zero constant term and Let hzi i∞
i=1 be a sequence in Z. Then there exists F ∈ Pf (N)
such that {a ∈ A : {a + p(Σi∈F zi ) : p ∈ R} ⊆ A} is piecewise syndetic.
Proof. (a) Corollary 2.9.
(b) [13, Lemma 14.24].
(c) [13, Theorem 15.16(b)]
(d) [12, Theorem 2.10].
(e) In [3, Theorem C] it was shown that the conclusion follows from the assumption
that A is piecewise syndetic. For an algebraic proof see [11, Corollary 3.7].
3.4 Lemma. Let D = {q ∈ ∆ : q is a minimal idempotent of (βN, +)}. Then c`D is
a left ideal of (βN, ·).
Proof. We have already observed that D 6= ∅. Let r ∈ c`D. To see that βN · r ⊆ c`R
it suffices by the continuity of ρr in (βN, ·) to show that N · r ⊆ c`D. So let x ∈ N and
let A ∈ x · r. Then x−1 A ∈ r so pick q ∈ D ∩ x−1 A. Then A ∈ x · q. By [13, Theorem
6.79] x · q ∈ ∆. By [12, Lemma 2.1] x · q is a minimal idempotent of (βN, +).
Plentiful examples of candidates for the sets G and H of Theorem 3.5 are provided
by Theorems 3.2 and 3.3. Notice in particular that H could be any family of subsets of
N such that any additively central set must contain a member of H.
3.5 Theorem. Let D = {q ∈ ∆ : q is a minimal idempotent of (βN, +)}. Let G be a
set of finite subsets of N with the property that any multiplicatively central subset of N
contains a member of G and let H be a set of (finite or infinite) subsets of N with the
property that, whenever A ⊆ N and A ∩ D 6= ∅, some member of H is contained in A.
Sr
Whenever r ∈ N and N = i=1 Ai , there exists i ∈ {1, 2, . . . , r} such that d(Ai ) > 0,
d∗m (Ai ) > 0, and there exist B ∈ G and C ∈ H such that B ∪ C ∪ B · C ⊆ Ai .
10
Proof. By Lemma 3.4, D is a left ideal of (βN, ·) so pick a minimal idempotent q
of (βN, ·) in c`D. Pick i ∈ {1, 2, . . . , r} such that Ai ∈ q. Since q ∈ c`D ⊆ ∆,
d(Ai ) > 0. By Theorem 3.2(b), d∗m (Ai ) > 0. Since q = q · q, {x ∈ Ai : x−1 Ai ∈ q} ∈ q.
In particular {x ∈ Ai : x−1 Ai ∈ q} is multiplicatively central, so pick B ∈ G such
T
that B ⊆ {x ∈ Ai : x−1 Ai ∈ q}. Since B is finite, Ai ∩ x∈B x−1 Ai ∈ q and thus
T
T
(Ai ∩ x∈B x−1 Ai ) ∩ D 6= ∅. Pick C ∈ H such that C ⊆ Ai ∩ x∈B x−1 Ai .
By adding the requirement that the members of H be finite, we obtain an infinitary
extension of Theorem 3.5 along the lines of the Central Sets Theorem.
3.6 Theorem. Let D = {q ∈ ∆ : q is a minimal idempotent of (βN, +)}. For each
n ∈ N, let Gn be a set of finite subsets of N with the property that any multiplicatively
central subset of N contains a member of Gn and let Hn be a set of finite subsets of N with
the property that, whenever A ⊆ N and A ∩ D 6= ∅, some member of Hn is contained in
Sr
A. Whenever r ∈ N and N = i=1 Ai , there exists i ∈ {1, 2, . . . , r} such that d(Ai ) > 0,
∞
d∗m (A) > 0, and there exist sequences hBn i∞
n=1 and hCn in=1 such that Bn ∈ Gn and
Cn ∈ Hn for each n and for any F ∈ Pf (N) and any f ∈ ×n∈F (Bn ∪ Cn ∪ Bn · Cn ),
Q
n∈F f (n) ∈ Ai .
Proof. Pick a minimal idempotent q of (βN, ·) in c`D and pick i ∈ {1, 2, . . . , r} such that
Ai ∈ q. Then d(Ai ) > 0 and d∗m (A) > 0. For any X ∈ q, let X ? = {x ∈ X : x−1 X ∈ q}.
Then by [13, Lemma 4.14] X ? ∈ q and for any x ∈ X ? , x−1 X ? ∈ q.
T
Choose B1 ∈ G1 such that B1 ⊆ A1 ? and choose C1 ∈ H1 such that C1 ⊆ A1 ? ∩
x∈B1
x−1 A1 ? .
Inductively, let n ∈ N and assume we have chosen Bt ∈ Gt and Ct ∈ Ht for each
t ∈ {1, 2, . . . , n} with the property that for all nonempty F ⊆ {1, 2, . . . , n} and all
Q
f ∈ ×t∈F (Bt ∪ Ct ∪ Bt · Ct ), t∈F f (t) ∈ Ai ? . Let
−1 ?
T Q
X = Ai ? ∩
Ai : ∅ 6= F ⊆ {1, 2, . . . , n} and
t∈F f (t)
f ∈ ×t∈F (Bt ∪ Ct ∪ Bt · Ct ) .
Then X is a finite intersection of members of q so X ∈ q. Pick Bn+1 ∈ Gn+1 such
T
that Bn+1 ⊆ X ? . Then X ∩ x∈Bn+1 x−1 X ∈ q so pick Cn+1 ∈ Hn+1 such that
T
Cn+1 ⊆ X ∩ x∈Bn+1 x−1 X.
We shall be concerned in the next section with extensions of the following sort of
configuration.
3.7 Corollary. Let m, k ∈ N and let N =
Sm
11
i=1
Ai . Then there exist i ∈ {1, 2, . . . , m},
a, d, b ∈ Ai , and r ∈ Ai \ {1} such that d(Ai ) > 0, d∗m (Ai ) > 0, and
s
br
:
s
∈
{0,
1,
.
.
.
,
k}
∪
a
+
td
:
t
∈
{0,
1,
.
.
.
,
k}
∪ {rd}
s
∪
r(a + td): t ∈ {0, 1, . . . , k} ∪ bdr : s ∈ {0, 1, . . . , k} ∪
br s (a + td) : s, t ∈ {0, 1, . . . , k} ⊆ Ai .
Proof. Let G = { br s : s ∈ {0, 1, . . . , k} ∪ {r} : b, r ∈ N} and let
H = { a + td : t ∈ {0, 1, . . . , k} ∪ {d} : a, d ∈ N} .
By applying Theorem 2.9 to (N, ·) and to (N, +) one concludes that every multiplicatively central set contains a member of G and that every additively central set contains
a member of H. Thus we may apply Theorem 3.6. By assigning 1 to its own cell one
may ensure that r 6= 1.
Sm
3.8 Corollary. Let m, k ∈ N and let N = i=1 Ai . Then there exist i ∈ {1, 2, . . . , m},
a, d ∈ Ai , and r ∈ Ai \ {1} such that d(Ai ) > 0, d∗m (Ai ) > 0, and
s
r (a + td) : s, t ∈ {0, 1, . . . , k} ∪ {dr s : s ∈ {0, 1, . . . , k}} ⊆ Ai .
Proof. Let i, a, b, d, r be as in the proof of Corollary 3.7. Put a1 = ab and d1 = db.
Then {a1 } ∪ {d1 } ∪ r s (a1 + td1 ) : s, t ∈ {0, 1, . . . , k} ∪ d1 r s : s ∈ {0, 1, . . . , k} ⊆ Ai .
.
4. Extensions of geoarithmetic progressions
A geoarithmetic progression is a set of the form r j (a + id) : i, j ∈ {0, 1, . . . , k} where
a, d, k ∈ N and r ∈ N \ {1}. We shall be concerned in this section with finding certain
generalizations of geoarithmetic progressions in one cell of a finite partition of N.
Our first result in this direction (Corollary 4.3) replaces r in a geometric progression
by multiples of members of any partition regular family of finite sets. For that result,
one needs to add a multiplier b because one can certainly not expect to find a set of the
form {r, r 2 } for r > 1 in one cell of an arbitrary finite partition of N; one may assign
the members of N \ x2 : x ∈ N \ {1} to A1 or A2 at will, and then assign x2 to the
cell that x is not in, x4 to the cell x2 is not in, and so on.
To establish Theorem 4.3 we need the following algebraic result which is of interest
in its own right. We let ω = N ∪ {0}. The case (S, +) = (ω, +) of Theorem 4.1 follows
from [12, Theorem 2.10]. In any semigroup S, a set C ⊆ S is central* if and only if for
every central subset B of S, C ∩ B 6= ∅. (Equivalently, S \ C is not central.) Notice in
12
particular that always S is central* so that if all first entries of a first entries matrix A
are equal to 1, the requirement that 1S be central* is automatically satisfied.
4.1 Theorem. Let u, v ∈ N and let A be a u × v first entries matrix with entries from
ω. Let (S, +) be a commutative semigroup with identity 0 and let C be a central subset
of S. If for every first entry c of A, cS is central*, then {~x ∈ S v : A~x ∈ C u } is central
in S v .
Proof. Pick a minimal idempotent e of βS such that C ∈ e. Define ϕ : S v → S u
e : β(S v ) → (βS)u be its continuous extension. Let M =
by ϕ(~x ) = A~x and let ϕ
e (p) = (e, e, . . . , e)T }. By [13, Corollary 4.22] ϕ
e is a homomorphism, so to
{p ∈ β(S v ) : ϕ
see that M is a subsemigroup, it suffices to show that M 6= ∅.
For each B ∈ e pick by [13, Theorem 15.5] ~xB ∈ S v such that ϕ(~xB ) ∈ B u . Direct
e by reverse inclusion and let q be a limit point in β(S v ) of the net h~xB iB∈e . Then
q ∈ M.
Since M is a compact right topological semigroup, pick a minimal idempotent r
of M . We claim that r is minimal in β(S v ). To see this, let p be an idempotent of
e (p) ≤ ϕ
e (r) = (e, e, . . . , e)T and (e, e, . . . , e)T is minimal
β(S v ) such that p ≤ r. Then ϕ
e (p) = (e, e, . . . , e)T . Thus p ∈ M and so p = r.
in (βS)u by [13, Theorem 2.23] so ϕ
e [ X ] ⊆ ( B )u . Then X ⊆ {~x ∈ S v : A~x ∈ B u }.
Pick X ∈ r such that ϕ
4.2 Theorem. Let (S, ·) be a commutative semigroup with identity and let C be a central
subset of S. If F is a partition regular family of finite subsets of S and k ∈ N, then there
exist b, r ∈ S and F ∈ F such that rF ∪ b(rx)j : x ∈ F and j ∈ {0, 1, . . . , k} ⊆ C.
Proof. Let k ∈ N and let

0
1

1
A=
.
 ..

1
0

1 .
.. 
.
1 k
Then A is a first entries matrix with all first entries equal to 1 so by Theorem 4.1
{(b, r) ∈ S 2 : {b, r, br, . . . , br k } ⊆ C} is central in S 2 and is in particular piecewise
syndetic. Let G = {b} × F : b ∈ S and F ∈ F . Then G is a partition regular family
of finite subsets of S 2 so pick by Lemma 2.3 F ∈ F , c ∈ S, and (s, r) ∈ S 2 such that
(s, r) · ({c} × F ) ⊆ {(b, r) ∈ S 2 : {b, r, br, . . . , br k } ⊆ C}. Let b = sc.
Notice that, if in the above proof, the matrix A is replaced by a matrix whose set
of rows is {(0, 0, 1)} ∪ (0, 1, j) : j ∈ {0, 1, . . . , k} ∪ (1, i, j) : i, j ∈ {0, 1, . . . , k} ,
13
then the conclusion of Theorem 4.2 becomes “there exist b, c, r ∈ S and F ∈ F such
that rF ∪ b(rx)j : x ∈ F and j ∈ {0, 1, . . . , k} ∪ cbi (rx)j : x ∈ F and i, j ∈ {0, 1,
. . . , k} ⊆ C.” Of course additional strengthenings can be obtained using first entries
matrices with all first entries equal to 1 and additional columns.
4.3 Corollary. Let F be a partition regular family of finite subsets of N, let k ∈ N,
and let A be piecewise syndetic in (N, ·). Then there exist b, r ∈ N and F ∈ F such that
{b(rx)j : j ∈ {0, 1, . . . , k} and x ∈ F } ⊆ A.
Proof. Pick by [13, Theorem 4.43] t ∈ N such that t−1 A is central in (N, ·). Pick by
Theorem 4.2 c, r ∈ N such that rF ∪ c(rx)j : x ∈ F and j ∈ {0, 1, . . . , k} ⊆ t−1 A and
let b = tc.
We see now that, given any central subset C of (N, ·) we can get sets of the form
b(a + id)j : i, j ∈ {0, 1, . . . , k} together with the multiplier, the increment, and the
arithmetic progression in C.
4.4 Corollary. Let C be a central subset of (N, ·) and let k ∈ N. There exist a, b, d ∈ N
such that
j
b(a + td)j : t, j ∈ {0, 1, . . . , k} ∪ bd
: j ∈ {0, 1, . . . , k}
∪ a + td : t ∈ {0, 1, . . . , k} ∪ {d} ⊆ C .
Proof. Let F = {d, a, a + d, . . . , a + kd} : a, d ∈ N}. Pick by Theorem 4.2 b, r ∈ S and
F ∈ F such that rF ∪ b(rx)j : x ∈ F and j ∈ {0, 1, . . . , k} ⊆ C. Pick c, s ∈ N such
that F = {c, s, s + c, . . . , s + kc}. Let d = rc and a = rs.
Again note that if the stronger version of Theorem 4.2 that we mentioned after its
proof is used, the conclusion of Corollary 4.4 becomes “There exist a, b, c, d ∈ N such
that
i j
j
cbi (a
+
td)
:
t,
i,
j
∈
{0,
1,
.
.
.
,
k}
∪
cb
d
:
i,
j
∈
{0,
1,
.
.
.
,
k}
∪ b(a + td)j : t, j ∈ {0, 1, . . . , k} ∪ bdj : j ∈ {0, 1, . . . , k}
∪ a + td : t ∈ {0, 1, . . . , k} ∪ {d} ⊆ C .”
We remark also that Corollary 4.4 could also be stated in terms of an arbitrary commutative ring with no change in proof.
The following result is stronger than Corollary 4.4. We state it separately because
its formulation is more involved and the proof requires more theoretical background.
4.5 Corollary. Let S be an infinite set with operations + and · such that (S, +) is a
commutative semigroup with identity 0, (S \ {0}, ·) is a commutative semigroup with
identity 1, and · distributes over +. Let C be a central subset of (S \ {0}, ·), let k ∈ N,
14
and let G be a finite subset of S \ {0}. Then there exist a, b, d ∈ C such that
b(a + di)j : i ∈ G and j ∈ {0, 1, . . . , k} ∪ bdj : j ∈ {0, 1, . . . , k}
∪ {a + di : i ∈ G} ⊆ C .
Proof. We observe first that S\{0} is central in (S, +). To see this, suppose instead that
0 is a minimal idempotent of (βS, +). Then by [13, Theorem 2.9] βS = 0 + βS = βS + 0
is a group and in particular (S, +) is cancellative. But then by [13, Theorem 4.36] βS \S
is an ideal of (βS, +) and so 0 ∈ βS \ S, a contradiction.
Let F = {a, d} ∪ {a + dj : j ∈ G} : a, d ∈ S . We claim that F ∩ P(S \ {0}) is
Sr
partition regular in S \{0}. So let r ∈ N and let S \{0} = i=1 Di . Pick i ∈ {1, 2, . . . , r}
∞
such that Di is central in (S, +). Let hdn i∞
n=1 be a sequence such that F S(hdn in=1 ) ⊆ Di .
Theorem 2.8 applied to the sequences hjdn i∞
n=1 for j ∈ G yields that there exist a ∈ Di
P
P
and F ∈ Pf (N) such that a + t∈F jdt ∈ Di for all j ∈ G. If we let d = t∈F dt we see
that {a, d} ∪ {a + dj : j ∈ G} ⊆ Di .
Pick by Theorem 4.2 b, r ∈ S \ {0} and F ∈ F ∩ P(S \ {0}) such that rF ∪ b(rx)j :
x ∈ F and j ∈ {0, 1, . . . , k} ⊆ C. Pick c, s ∈ S such that F = {c, s} ∪ {s + ic : i ∈ G}.
Let d = rc and a = rs. Since a, d ∈ rF , we have a, d ∈ C. Also b = ba0 so b ∈ C.
Suppose that the semigroup S satisfies the hypotheses of Corollary 4.5 and that
0·x = 0 for every x ∈ S. Then, by [4, Theorem 4.4] first entry matrices over S whose first
entries are all 1, can be used to prove Corollary 4.5 as well as a sequence of successively
stronger theorems. For example, the theorem stated in the remark following Theorem
4.2 is valid in S if C is any central subset of (S \ {0}, ·), G is any given finite subset of
S and F = {f } ∪ {d + tf : t ∈ G} ∪ {a + sd + tf : s, t ∈ G} for some a, d, and f in
S \ {0}.
The following corollary is also a consequence of [1, Theorem 3.15].
4.6 Corollary. Let k ∈ N, and let A be piecewise syndetic in (N, ·). Then there exist
a, b, d ∈ N such that b(a + id)j : i, j ∈ {0, 1, . . . , k} ⊆ A.
Proof. Pick t ∈ N such that t−1 A is central and apply Corollary 4.4.
Now, as we promised in the introduction, we turn our attention to extensions of
the following result from [1].
∞
4.7 Theorem. Let m, k ∈ N. For each i ∈ {0, 1, . . . , k} let hxi,t i∞
t=1 and hyi,t it=1 be
Sm
sequences in N. Let N = s=1 As . Then there exist s ∈ {1, 2, . . . , m}, F, G ∈ Pf (N),
P
Q
and a, b ∈ N such that b(a + t∈F xi,t ) · ( t∈G yj,t ) : i, j ∈ {0, 1, . . . , k} ⊆ As .
15
Proof. By [1, Theorem 3.13], every set A with d∗m (A) > 0 contains such a configuration
and for some s, d∗m (As ) > 0.
We shall show in Theorem 4.8 that one may take F = G in Theorem 4.7 and in
Corollary 4.12(a) that the multiplier b may be eliminated. We show in Corollary 4.16,
however, that one cannot simultaneously take F = G and eliminate b.
∞
4.8 Theorem. Let m, k ∈ N. For each i ∈ {0, 1, . . . , k} let hxi,t i∞
t=1 and hyi,t it=1 be
Sm
sequences in N. Let N = s=1 As . Then there exist s ∈ {1, 2, . . . , m}, F ∈ Pf (N), and
a, b ∈ N such that
P
{ba}
∪
b(a
+
x
)
:
i
∈
{0,
1,
.
.
.
,
k}
∪
i,t
t∈F
Q
1, . . . , k} ∪
t∈F yj,t : j ∈ {0,
ba · P
Q
b(a + t∈F xi,t ) · ( t∈F yj,t ) : i, j ∈ {0, 1, . . . , k} ⊆ As .
Proof. Let xk+1,t = 0 and yk+1,t = 1 for all t. Let A0 = {0}. Let L = {0, 1, . . . ,
k + 1} × {0, 1, . . . , k + 1} and let S be the free semigroup on the alphabet L. Given
Pn
Qn
a word w = l1 l2 · · · ln of length n in S, define f (w) =
x
·
π
(l
),t
1 t
t=1
t=1 yπ2 (lt ),t .
Sm −1
Then S = s=0 f [As ] so pick by Theorem 2.7, s ∈ {0, 1, . . . , m} and a variable word
w = l1 l2 · · · ln (with each lt ∈ L ∪ {v}) such that {w(c) : c ∈ L} ⊆ As . Notice that
s 6= 0.
Let F = {t ∈ {1, 2, . . . , n} : lt = v} and let G = {1, 2, . . . , n} \ F . Let a =
Q
t∈G xπ1 (lt ),t and let b =
t∈G yπ2 (lt ),t . Then given i, j ∈ {0, 1, . . . , k+1}, f w (i, j)
P
Q
= (a + t∈F xi,t ) · b · t∈F yj,t .
P
∞
4.9 Corollary. Let k ∈ N. For each i ∈ {0, 1, . . . , k} let hxi,t i∞
t=1 and hyi,t it=1 be
sequences in N and let A be piecewise syndetic in (N, ·). Then there exist F ∈ P f (N)
and a, b ∈ N such that
P
{ba} ∪ b(a + t∈F xi,t ) : i ∈ {0, 1, . . . , k} ∪
Q
yj,t : j ∈ {0, 1, . . . , k} ∪
ba t∈F
P
Q
b(a + t∈F xi,t ) · ( t∈F yj,t ) : i, j ∈ {0, 1, . . . , k} ⊆ A .
Proof. By Theorem 4.8 the collection of sets H of the form
P
H = {ba}
∪
b(a
+
x
)
:
i
∈
{0,
1,
.
.
.
,
k}
∪
i,t
t∈F
Q
y : j ∈ {0, 1, . . . , k} ∪
ba t∈F
P j,t
Q
b(a + t∈F xi,t ) · ( t∈F yj,t ) : i, j ∈ {0, 1, . . . , k}
is partition regular, so by Lemma 2.3 there is some t ∈ N and some such H with tH ⊆ A.
Replacing b by tb yields the desired conclusion.
4.10 Lemma. Let (S, ·) be a commutative semigroup, let L be a minimal left ideal of
(βS, ·), and let k ∈ N. Let F be a family of finite subsets of S such that the family
16
{bF : F ∈ F and b ∈ S} is partition regular. Let A ⊆ S such that A ∩ L 6= ∅. Then
T
there exists F ∈ F such that L ∩ y∈F y −1 A 6= ∅.
Proof. Pick v ∈ A ∩ L. Pick a minimal right ideal R of (βS, ·) such that v ∈ R and
pick an idempotent u ∈ R. Then v = uv so B = {x ∈ S : x−1 A ∈ v} ∈ u. In particular
B is central so pick by Lemma 2.3, some b ∈ S and F ∈ F such that bF ⊆ B. So for
each y ∈ F , (by)−1 A ∈ v. Equivalently for each y ∈ F , y −1 A ∈ bv. Since bv ∈ L, we
are done.
We have by Lemma 3.4 that if D = {q ∈ ∆ : q is a minimal idempotent of
(βN, +)} then c`D is a left ideal of (βN, ·) and consequently c`D ∩ K(βN, ·) 6= ∅. Given
any p ∈ βS and any finite partition {A1 , . . . , Am } there is at least one cell Ai such that
Ai ∈ p. Consequently, the partition versions of Theorem 4.11 and Corollary 4.12 are
also valid.
4.11 Theorem. Let D = {q ∈ ∆ : q is a minimal idempotent of (βN, +)} and let A
be a subset of N such that A ∩ c`D ∩ K(βN, ·) 6= ∅. Let F be a family of finite subsets of
N such that the family {bF : F ∈ F and b ∈ N} is partition regular and let G be a family
of subsets of N such that any set which is central in (N, +) contains a member of G.
T
T
Then there exist F ∈ F and G ∈ G such that d( y∈F y −1 A) > 0, d∗m ( y∈F y −1 A) > 0
and F G ⊆ A.
Proof. Pick a minimal left ideal L of (βN, ·) such that A∩c`D∩L 6= ∅. Since c`D is a left
T
ideal of (βN, ·), L ⊆ c`D. Pick, by Lemma 4.10, F ∈ F such that L ∩ y∈F y −1 A 6= ∅.
T
Since L ⊆ K(βN, ·) ⊆ ∆∗m by [13, Theorems 20.5 and 20.6] d∗m ( y∈F y −1 A) > 0. Since
T
L ⊆ c`D, pick q ∈ ∆ such that q is a minimal idempotent of (βN, +) and y∈F y −1 A ∈ q.
T
Then this set is central in (N, +) so pick G ∈ G such that G ⊆ y∈F y −1 A. Since q ∈ ∆,
T
d( y∈F y −1 A) > 0.
4.12 Corollary. Let D = {q ∈ ∆ : q is a minimal idempotent of (βN, +)}, let A be a
subset of N such that there is a multiplicative idempotent p ∈ A ∩ c`D ∩ K(βN, ·), and
let k ∈ N.
∞
(a) For each i ∈ {1, 2, . . . , k} let hxi,t i∞
t=1 and hyi,t it=1 be sequences in N. Then
Tk Q
there exist H, K ∈ Pf (N) and a ∈ A such that d(A ∩ j=1 ( t∈H yj,t )−1 A) > 0,
Tk Q
d∗m (A ∩ j=1 ( t∈H yj,t )−1 A) > 0, and
Q
P
a + t∈K
x
:
i
∈
{1,
2,
.
.
.
,
k}
∪
a
·
y
:
j
∈
{1,
2,
.
.
.
,
k}
∪
i,t
j,t
t∈H
P
Q
(a + t∈K xi,t ) · t∈H yj,t : i, j ∈ {1, 2, . . . , k} ⊆ A .
17
Tk
Tk
(b) There exist a, r, d ∈ A such that r > 1, d( j=0 (r j )−1 A) > 0, d∗m ( j=0 (r j )−1 A) >
0, and (a + id)r j : i, j ∈ {0, 1, . . . , k} ∪ dr j : j ∈ {0, 1, . . . , k} ⊆ A.
Tk
Tk
(c) There exist a, r, d ∈ A such that r > 1, d( j=1 (j r )−1 A) > 0, d∗m ( j=0 (j r )−1 A) >
0, and dj r : j ∈ {1, 2, . . . , k} ∪ (a + id)j r : i ∈ {0, 1, . . . , k} and j ∈ {1, 2, . . . ,
k} ∪ a + id : i ∈ {0, 1, . . . , k} ⊆ A.
Proof. Since 1 is not an element of any minimal left ideal of (βN, ·), by considering
A \ {1} instead of A we may assume that 1 ∈
/ A. Let
Q
F1 = {1} ∪
t∈H yi,t : i ∈ {1, 2, . . . , k} : H ∈ Pf (N) ,
P
G1 = {a} ∪ {a + t∈K xi,t : i ∈ {1, 2, . . . , k} : K ∈ Pf (N) and a ∈ N ,
F2 = { r i : i ∈ {0, 1, . . . , k} : r ∈ A} ,
G2 = {d} ∪ {a + id : i ∈ {0, 1, . . . , k}} : a, d ∈ N ,
and put Fi0 = {bF : b ∈ N and F ∈ Fi } for i ∈ {1, 2}. By applying Theorem 2.9 and
Corollary 2.10 to the semigroup (N, ·) we see that the families F10 and F20 are partition
regular. Similarly by Theorem 2.9 and Corollary 2.10 applied to the semigroup (N, +),
every subset of N that is central in (N, +) contains a member of G1 and a member of G2 .
Thus we get (a) by applying Theorem 4.11 to F1 and G1 and (b) by applying Theorem
4.11 to F2 and G2 .
We will prove (c) by using Theorem 4.11 with F1 and G2 , where we define the
sequences hyi,n i∞
n=1 , i ∈ {1, 2, . . . , k} appropriately. Since A is central in (N, +), choose
rn
∞
for i ∈ {1, 2,
a sequence hrn in=1 such that F S(hrn i∞
n=1 ) ⊆ A. Using this put yi,n = i
. . . , k} and n ∈ N. By Theorem 4.11 we find a, d ∈ A and H ∈ Pf (N) such that
Q
G = {d} ∪ {a + id : i ∈ {0, 1, . . . , k}} and F = {1} ∪
t∈H yj,t : j ∈ {1, 2, . . . , k}
P
satisfy the conclusion of Theorem 4.11. Let r = t∈H rt ∈ A. Then for j ∈ {1, 2, . . . , k},
Q
Q
rt
= j r . Thus we see that (c) is valid.
t∈H yj,t =
t∈H j
We now turn our attention to showing that one cannot simultaneously let F = G
and eliminate the multiplier b in Theorem 4.7.
The following theorem is of interest in its own right. Recall from Corollary 2.9
that when N is finitely colored, one can find arbitrarily long monochrome arithmetic
progressions with increments chosen from any IP-set. This theorem tells us that at least
relatively thin sequences cannot replace IP-sets.
4.13 Theorem. Let hdn i∞
n=1 be a sequence in N such that for all n ∈ N, 3d n ≤ dn+1 .
There exists a partition {A0 , A1 , A2 , A3 } of N such that there do not exist s ∈ {0, 1, 2, 3}
and a, k ∈ N with {a, a + dk } ⊆ As .
18
Proof. For α ∈ T = R/Z we denote by kαk the distance to the nearest integer. We will
not distinguish strictly between equivalance classes and their representatives in [0, 1).
4.14 Lemma. There exists α ∈ T such that kαdn k ≥ 1/4 for each n ∈ N.
Proof. For each n ∈ N put Rn = {α ∈ T : kαdn k ≥ 1/4}. Each Rn consists of intervals
1
of length
which are separated by gaps of the same length. Since dn+1 ≥ 3dn every
2dn
interval of Rn is 3 times longer than an interval or a gap of Rn+1 . Thus any interval of
TN
Rn contains an interval of Rn+1 . This shows that for each N ∈ N, n=1 Rn 6= ∅. By
T∞
compactness of T there exists α ∈ n=1 Rn .
Let α ∈ T such that dn α ∈ [1/4, 3/4] for each n ∈ N. For i ∈ {0, 1, 2, 3} put
Ai = {m ∈ N : mα ∈ [i/4, (i + 1)/4)}. Then for any a, k ∈ N α(a + dk ) = αa + β for
some β ∈ [1/4, 3/4] and thus αa and α(a + dk ) must not lie in the same quarter of [0, 1).
Equivalently there exists no s ∈ {0, 1, 2, 3} such that {a, a + dk } ⊆ As .
We remark that Lemma 4.14 is well known. Under the much weaker assumption,
that the growth rate of the sequence hdn i∞
n=1 is bounded from below by some q > 1
B. de Mathan [14] and A. Pollington [15] independently proved that there exists some
α ∈ T such that {αn : n ∈ N} is not dense in T. In order to give a self contained proof
we have chosen to go with the weaker statement. The loss is that we have to make an
additional step to show that any growth rate q > 1 is sufficient to avoid monochrome
arithmetic progressions with some dk as increment.
4.15 Corollary. Let q ∈ R with q > 1 and assume that hdn i∞
n=1 is a sequence in N
such that for all n ∈ N, qdn ≤ dn+1 . There exists a finite partition {A1 , A2 , . . . , Ar } of
N such that there do not exist s ∈ {1, 2, . . . , r} and a, k ∈ N with {a, a + d k } ⊆ As .
Proof. Pick m ∈ N such that q m ≥ 3. For t ∈ {0, 1, . . . , m − 1} and n ∈ N, let
ct,n = dnm−t . Given t ∈ {0, 1, . . . , m} one has that 3ct,n ≤ ct,n+1 for each n so pick by
Theorem 4.13 some {Bt,0 , Bt,1 , Bt,2 , Bt,3 } of N such that there do not exist s ∈ {0, 1, 2, 3}
and a, k ∈ N with {a, a+ct,k } ⊆ Bt,i . Let r = 4m and define a partition {A1 , A2 , . . . , Ar }
of N with the property that x and y lie in the same cell of the partition if and only if
x ∈ Bt,i ⇔ y ∈ Bt,i for each t ∈ {0, 1, . . . , m − 1} and each i ∈ {0, 1, 2, 3}.
∞
∞
4.16 Corollary. There exist sequences hx0,n i∞
n=1 , hx1,n in=1 , and hyn in=1 in N and a
partition {A0 , A1 , A2 , A3 } of N such that there do not exist s ∈ {0, 1, 2, 3}, F ∈ Pf (N),
P
Q
and a ∈ N with (a + n∈F xi,n ) · n∈F yn : i ∈ {0, 1} ⊆ As .
19
Proof. For each t ∈ N, let x0,t = 1, x1,t = 2 and yi,t = 3. For each n ∈ N, let dn = n3n .
Pick A0 , A1 , A2 , A3 as guaranteed by Theorem 4.13. Suppose one has F ∈ Pf (N) and
P
Q
a ∈ N with (a + t∈F xi,t ) · t∈F yt : i ∈ {0, 1} ⊆ As . Let n = |F |. Then
P
Q
P
Q
(a + t∈F x1,t ) · t∈F yt = dn + (a + t∈F x0,t ) · t∈F yt , a contradiction.
We have just shown that one cannot eliminate the multiplier b from Theorem 4.8.
We show now that this multiplier cannot be eliminated from Corollary 4.9. Recall that
thick sets in any semigroup are also piecewise syndetic, in fact central.
4.17 Theorem. There exists a set A which is thick in (N, ·) and a sequence hx n i∞
n=1 in
N with the property that there do not exist a ∈ N and d ∈ F S(hxn i∞
n=1 ) with {a, a + d} ⊆
A.
S∞
Proof. Let A = n=1 {(3n)!, 2(3n)!, . . . , n(3n)!} and for each n, let xn = (3n + 1)! .
Observe that A is thick in (N, ·). Let a ∈ A and let d ∈ F S(hxn i∞
n=1 ). We shall show
that a + d ∈
/ A. Pick n ∈ N and k ∈ {1, 2, . . . , n} such that a = k(3n)! . Pick F ∈ Pf (N)
P
such that d = t∈F xt and let m = max F . Then (3m + 1)! ≤ d < (3m + 2)!.
If m < n we have k(3n)! < a + d < (k + 1)(3n)! so a + d ∈
/ A. If m ≥ n, then
a < (3m + 1)! so (3m + 1)! < a + d < (3m + 3)! and thus a + d ∈
/ A.
It was shown in [1, Theorem 1.3] that the fact that a subset A of N satisfies d∗m (A) >
0 is enough to guarantee that A contains arbitrarily large geoarithmetic progressions.
However, by considering the set A = {x ∈ N : the number of terms in the prime
factorization of x is odd}, one sees that the fact that dm (A) > 0 is not enough to
guarantee geoarithmetic progressions together with the common ratio r, nor together
with both b and a.
As is well known among afficianados, geoarithmetic progressions are strongly partition regular . That is, for each m, k ∈ N there exists K ∈ N such that whenever
Sm
A, B, D ∈ N, R ∈ N \ {1}, and BRs (A + tD) : s, t ∈ {0, 1, . . . , K} = i=1 Ci , there
exist i ∈ {1, 2, . . . , m}, a, b, d ∈ N, and r ∈ N \ {1} such that br s (a + td) : s, t ∈ {0, 1,
. . . , k} ⊆ Ai . (The easiest way to see this is to use the Grünwald/Gallai Theorem1
[8, Theorem 2.8]. Color the pair (s, t) ∈ {0, 1, . . . , K} × {0, 1, . . . , K} according to the
color of BRs (A + tD).)
We show now that configurations of the sort produced by Corollary 3.7 are not
strongly partition regular.
1 This theorem was never published by its author. Its first publication was in [16] where it was
referred to as Grünwald’s Theorem, Grünwald being the original name of the author. During the period
surrounding World War II Grünwald changed his name to Gallai.
20
4.18 Theorem. There is a set C ⊆ N such that for each k ∈ N there exist b, a, d ∈ N
and r ∈ N \ {1} such that br n (a + td) : n, t ∈ {0, 1, . . . , k} ∪ br n : n ∈ {0, 1,
. . . , k} ∪ a + td : t ∈ {0, 1, . . . , k} ⊆ C and there exist sets A1 and A2 such that
C = A1 ∪ A2 and there do not exist i ∈ {1, 2}, c, a, d ∈ N, and s ∈ N \ {1} such that
{cs, cs2 , cs(a + d), cs2 (a + d), cs(a + 2d)} ⊆ Ai .
2
Proof. Let r1 = 5. Inductively choose a prime rk+1 > rk (2k + 1) . For each k ∈ N,
let Bk = rk n x : n ∈ {1, 2, . . . , k + 1} and x ∈ {k + 1, k + 2, . . . , 2k + 1} and let
S∞
B = k=1 Bk .
4.19 Lemma. If a, d ∈ N and {a + d, a + 2d} ⊆ B, then there exist k ∈ N and n ∈ {1, 2,
. . . , k + 1} such that {a + d, a + 2d} ⊆ rk n x : x ∈ {k + 1, k + 2, . . . , 2k + 1} .
Proof. Pick k ∈ N, n ∈ {1, 2, . . . , k + 1}, and x ∈ {k + 1, k + 2, . . . , 2k + 1} such that
a + d = rk n x. Then a + 2d < 2(a + d) = 2rk n x. Also 2rk n x < rk n+1 (k + 1) and
2rk n x < rk+1 (k + 2). The first member of B larger than rk n (2k + 1) is rk n+1 (k + 1) (if
n ≤ k) or rk+1 (k + 2) (if n = k + 1). Thus there is some y ∈ {x + 1, x + 2, . . . , k + 1}
such that a + 2d = rk n y.
4.20 Lemma. If c ∈ N, s ∈ N \ {1}, and {cs, cs2 } ⊆ B, then there exist k ∈ N,
n ∈ {0, 1, . . . , k}, t ∈ {1, 2, . . . , k + 1 − n}, and y ∈ {k + 1, k + 2, . . . , 2k + 1} such that
c = rk n y and s = rk t .
Proof. Pick k ≤ m, δ ∈ {1, 2, . . . , k +1}, ν ∈ {1, 2, . . . , m+1}, y ∈ {k +1, k +2, . . . , 2k +
1}, and z ∈ {m + 1, m + 2, . . . , 2m + 1} such that cs = rk δ y, and cs2 = rm ν z.
rm ν z
rm
Now s ≤ rk δ y ≤ rk k+1 (2k + 1) and s =
>
so
rk δ y
rk k+1 (2k + 1)
2
rm < rk k+1 (2k + 1) < rk+1
z
and so m ≤ k and thus m = k. Therefore s = rk ν−δ . Since rk is a prime which does
y
not divide y, we must have that y divides z and therefore that y = z. Let t = ν − δ.
Since crk ν−δ = cs = rk δ y we have c = rk 2δ−ν y. Let n = 2δ − ν. Since c = rk n y and
s = rk t we have that n ≥ 0 and t ≥ 1. Since n + t = δ we have that n + t ≤ k + 1.
To complete the proof of the theorem, let A1 = B, let A2 = rk n : k ∈ N and n ∈
{1, 2, . . . , k + 1} , and let C = A1 ∪ A2 . Given k ∈ N, let a = rk (k + 1) and let
d = b = r = rk . Then for t, n ∈ {0, 1, . . . , k − 1} one has br n = rk n+1 ∈ A2 , a + td =
rk (k + t + 1) ∈ A1 , and br n (a + td) = rk n+2 (k + t + 1) ∈ A1 .
21
It is trivial that A2 does not contain {cs(a + d), cs(a + 2d)} as the latter element
is less than twice the former. Suppose we have some c, a, d ∈ N and some s ∈ N \ {1}
such that
{cs, cs2 , cs(a + d), cs2 (a + d), cs(a + 2d)} ⊆ A1 .
Pick by Lemma 4.20 some k ∈ N, n ∈ {0, 1, . . . , k}, t ∈ {1, 2, . . . , k + 1 − n},
and y ∈ {k + 1, k + 2, . . . , 2k + 1} such that c = rk n y and s = rk t . Again invoking
Lemma 4.20, pick some k 0 ∈ N, m ∈ {0, 1, . . . , k 0 }, t0 ∈ {1, 2, . . . , k 0 + 1 − m}, and
0
z ∈ {k 0 + 1, k 0 + 2, . . . , 2k 0 + 1} such that c(a + d) = rk0 m z and s = rk0 t .
0
Since rk0 t = s = rk t we have k = k 0 and t = t0 . Pick by Lemma 4.19 k 00 ∈ N and
ν ∈ {1, 2, . . . , k 00 + 1} such that
{cs(a + d), cs(a + 2d)} ⊆ rk00 ν w : w ∈ {k 00 + 1, k 00 + 2, . . . , 2k 00 + 1} .
Since cs(a + d) = rk t+m z we have k 00 = k and ν = t + m. Since cs = rk t+n y we have
z
a + d = rk m−n . Since rk is a prime which does not divide y we have that y divides z
y
so y = z and thus a + d = rk m−n .
Pick w ∈ {k + 1, k + 2, . . . , 2k + 1} such that cs(a + 2d) = rk t+m w. Then a + 2d =
w
rk m−n so w divides y and thus a + 2d = rk m−n . Therefore d = 0, a contradiction.
y
We now present a general result which is strong enough to establish an extension
of Theorem 1.4.
4.21 Theorem. Let (S, ·) be a semigroup, let F be a set of subsets of S with the property
that each central subset of S contains a member of F , let G be a partition regular family
of finite subsets of S, and let A be a central subset of S. Then there exist F ∈ F , G ∈ G,
and t ∈ S such that F ∪ tGF ⊆ A.
Proof. Pick a minimal idempotent p of βS with A ∈ p. Then by [13, Theorem 4.39]
{s ∈ S : s−1 A ∈ p} is syndetic so pick H ∈ Pf (S) such that
S
S = t∈H t−1 {s ∈ S : s−1 A ∈ p} .
Pick G ∈ G and t ∈ H such that G ⊆ t−1 {s ∈ S : s−1 A ∈ p}. Then for each s ∈ G,
T
T
(ts)−1 A ∈ p so A ∩ s∈G (ts)−1 A ∈ p. Pick F ∈ F such that F ⊆ A ∩ s∈G (ts)−1 A.
4.22 Corollary. Let (S, ·) be a semigroup, let F and G be partition regular families of
finite subsets of S. Assume that for all F ∈ F and all x, t ∈ S, tF x ∈ F and let A be
a piecewise syndetic subset of S. Then there exist F ∈ F , G ∈ G, and t ∈ S such that
tGF ⊆ A. If S is commutative, then there exist F ∈ F and G ∈ G such that GF ⊆ A.
22
Proof. Note that by Lemma 2.3 F has the property that every piecewise syndetic
subset of S contains a member of F . In particular every central subset of S contains a
member of F . Pick by [13, Theorem 4.43] some x ∈ S such that x−1 A is central. Pick
by Theorem 4.21 some F ∈ F , G ∈ G, and t ∈ S such that F ∪ tGF ⊆ x−1 A. Then
(xt)GF ⊆ A.
The following corollary extends Theorem 1.4. Recall that any central set is a
piecewise syndetic IP set.
4.23 Corollary. Let A be a piecewise syndetic IP-set in (N, ·) with 1 ∈
/ A and let k ∈ N.
Then there exist a, r, d ∈ A such that
j
r (a + id) : i, j ∈ {0, 1, . . . , k} ∪ dr j : j ∈ {0, 1, . . . , k} ⊆ A .
Proof. Let F = { br j : j ∈ {0, 1, . . . , k} : b ∈ N and r ∈ A} and let
G = {d} ∪ a + id : i ∈ {0, 1, . . . , k} : a, d ∈ N .
By Corollary 2.9, F and G are partition regular. And trivially if F ∈ F and t ∈ N,
then tF ∈ F . Pick by Corollary 4.22 some F ∈ F and G ∈ G such that GF ⊆ A. Pick
b ∈ N and r ∈ A such that F = br j : j ∈ {0, 1, . . . , k} and pick a1 , d1 ∈ N such that
G = {d1 } ∪ a1 + id1 : i ∈ {0, 1, . . . , k} . Let a = a1 b and d = d1 b.
We see that we can turn the tables somewhat, translating geometric progressions
by arithmetic progressions. (Since addition does not distribute over multiplication, we
end up with the four variables a, d, b, and r, rather than just the three of Corollary
4.23.)
4.24 Corollary. Let A be a piecewise syndetic IP-set in (N, +) and let k ∈ N. Then
there exist d ∈ A, a, b ∈ N, and r ∈ N \ {1} such that
a + id + br j : i, j ∈ {0, 1, . . . , k} ∪ a + id + r : i ∈ {0, 1, . . . , k} ⊆ A .
Proof. Let F = { a + id : i ∈ {0, 1, . . . , k} : a ∈ N and d ∈ A} and let
G = {r} ∪ br j : j ∈ {0, 1, . . . , k} : b ∈ N and r ∈ N \ {1} .
Exactly as in the proof of Corollary 4.23, F and G are partition regular and if F ∈ F
and t ∈ N, then t + F ∈ F . Pick by Corollary 4.22 F ∈ F and G ∈ G such that
G + F ⊆ A. Pick b ∈ N and r ∈ N \ {1} such that G = {r} ∪ br j : j ∈ {0, 1, . . . , k} .
Pick a ∈ N and d ∈ A such that F = a + id : i ∈ {0, 1, . . . , k} .
We do not know whether we can require that any of a, b, or r be in A in Corollary
4.24.
23
5. Algebra in (β N, +) and (β N,·)
– extending the Central Sets Theorem
In attempting to derive results about geoarithmetic progressions, the approach that one
might try first after a little experience in deriving Ramsey Theoretic consequences of
the algebra of βN would be to choose an appropriate idempotent q in (βN, ·) and show
Tk
that if A ∈ q, then there is some r, preferably in A, such that s=0 (r s )−1 A ∈ q. We
show first that such an approach is doomed to failure.
5.1 Theorem.
(a) For all q ∈ βN, there exists a partition {A0 , A1 } of N such that for all i ∈ {0, 1}
and all x ∈ N, (−x + Ai ) ∩ (−2x + Ai ) ∈
/ q. In particular there exists A ∈ q such
that for all x ∈ N, either −x + A ∈
/ q or −2x + A ∈
/ q.
(b) There does not exist q ∈ βN such that for each A ∈ q there is some r ∈ N \ {1}
with r −1 A ∈ q and (r 2 )−1 A ∈ q.
Proof. (a) Let q ∈ βN. Then q + βN is a right ideal of (βN, +) so there is an additive
idempotent in q + βN. Pick r ∈ βN such that q + r is an idempotent in (βN, +). Then
T∞
q + r ∈ n=1 c`(N2n ) by [13, Lemma 6.6].
P
Define f : N → ω by f (n) = min F where F ∈ Pf (ω) and n = t∈F 2t . Then f
has a continuous extension fe : βN → βω. For i ∈ {0, 1} let Ai = {x ∈ N : (2N − i) ∈
fe(x + r)}.
Let i ∈ {0, 1} and let x ∈ N and suppose that (−x + Ai ) ∩ (−2x + Ai ) ∈ q. Pick
j, k ∈ ω such that x = 2j (2k + 1). Denote addition of z on the left in βN by λz and
addition of z on the right by ρz . Then fe ◦ λx is constantly equal to f (x) and fe ◦ λ2x is
constantly equal to f (x)+1 on N2j+2 , which is a member of q +r. So fe(x+q +r) = f (x)
and fe(2x+q +r) = f (x)+1. Therefore fe◦λx ◦ρr (q) = f (x) and fe◦λ2x ◦ρr (q) = f (x)+1
so
{y ∈ N : fe(x + y + r) = f (x) and fe(2x + y + r) = f (x) + 1} ∈ q
so pick y ∈ (−x + Ai ) ∩ (−2x + Ai ) such that fe(x + y + r) = f (x) and fe(2x + y + r) =
f (x) + 1.
Since x + y ∈ Ai , we have that 2N − i ∈ fe(x + y + r) = f (x) so f (x) + i ∈ 2N.
(Recall that we are identifying points of N with the principle ultrfilters they generate.)
Since 2x + y ∈ Ai , we have that 2N − i ∈ fe(2x + y + r) = f (x) + 1 so f (x) + i + 1 ∈ 2N,
a contradiction.
24
(b) For x ∈ N \ {1}, let `(x) be the number of terms in the prime factorization of x.
Then ` is a homomorphism from (N \ {1}, ·) onto (N, +) and so its continuous extension
è : (βN \ {1}, ·) → (βN, +) is also a homomorphism by [13, Corollary 4.22].
We know that there exist multiplicative idempotents in the closure of the set of
additive idempotents in βN. In fact, there exist minimal multiplicative idempotents in
the closure of the set of minimal additive idempotents in βN, and we used one such in
the proof of Theorem 3.6. In particular we know that c`K(βN, +) ∩ K(βN, ·) 6= ∅. In
the following we shall assume that geometric progressions have integer common ratios,
though the lemma would remain valid with the more liberal definition.
5.2 Lemma. Let D = {q ∈ βN : for all A ∈ q, A contains arbitrarily long geometric
progressions}. Then D is a closed two sided ideal of (βN, ·). In particular c`K(βN, ·) ⊆
D.
Proof. Trivially D is closed. Let q ∈ D, let s ∈ βN, let A ∈ qs, and let B ∈ sq. Let
n ∈ N. We need to show that A and B contain length n geometric progressions. Now
{x ∈ N : x−1 A ∈ s} ∈ q so pick a ∈ N and r ∈ N\{1} such that {a, ar, ar 2, . . . , ar n−1 } ⊆
Tn−1
Tn−1
{x ∈ N : x−1 A ∈ s}. Then t=0 (ar t )−1 A ∈ s so pick b ∈ t=0 (ar t )−1 A. Then
{ba, bar, bar 2, . . . , bar n−1 } ⊆ A. Also {x ∈ N : x−1 B ∈ q} ∈ s so pick x ∈ N such that
x−1 B ∈ q. Pick c ∈ N and d ∈ N \ {1} such that {c, cd, cd2 , . . . , cdn−1 } ⊆ x−1 B. Then
{xc, xcd, xcd2 , . . . , xcdn−1 } ⊆ B.
We see now that there would be interesting Ramsey theoretic consequences of the
existence of an additive idempotent in the set D defined above. (Compare the conclusion
with those of Theorem 3.6.)
5.3 Theorem. Let D = {q ∈ βN : for all A ∈ q, A contains arbitrarily long geometric
progressions} and assume that there exists q ∈ D such that q + q = q. Then whenever
Sr
r ∈ N and N = i=1 Ai , there exist i ∈ {1, 2, . . . , r} and a sequence hHn i∞
n=1 such that
for each n ∈ N, Hn is a length n geometric progression and for every F ∈ Pf (N), one
P
has n∈F Hn ⊆ Ai .
Proof. Pick q ∈ D such that q + q = q. Given B ∈ q, let B ? = {x ∈ B : −x + B ∈ q}.
Then by [13, Lemma 4.14], whenever x ∈ B ? one has −x + B ? ∈ q.
Pick i ∈ {1, 2, . . . , r} such that Ai ∈ q. Pick x ∈ Ai ? and let H1 = {x}. Let n ∈ N
and assume that hHt int=1 have been chosen so that for any F with ∅ 6= F ⊆ {1, 2, . . . , n}
P
and any f ∈ ×t∈F Ht , t∈F f (t) ∈ Ai ? . Let
T P
B = Ai ? ∩ {− t∈F f (t) + Ai ? : F ∈ Pf ({1, 2, . . . , n}) and f ∈ ×t∈F Ht } .
25
Then B ∈ q so pick a length n + 1 geometric progression Hn+1 ⊆ B.
We now turn our attention to deriving an extension, Theorem 5.8, of the Central
Sets Theorem for countable commutative semigroups [13, Theorem 14.11]. The Central
Sets Theorem for (N, +) is due to Furstenberg [6, Proposition 8.21]. See [13, Part III]
for numerous combinatorial applications of the Central Sets Theorem. Theorem 5.8 has
several earlier theorems as immediate corollaries. In particular, it implies a stronger
version of Theorem 4.8. To establish this theorem we shall use the notion of partial
semigroup introduced in [2].
5.4 Definition.
(a) A partial semigroup is a set S together with an operation · that maps a subset of
S × S into S and satisfies the associative law (x · y) · z = x · (y · z) in the sense that
if either side is defined, then so is the other and they are equal.
(b) Given a partial semigroup (S, ·) and x ∈ S, ϕ(x) = {y ∈ S : x · y is defined}.
(c) Given a partial semigroup (S, ·), x ∈ S, and A ⊆ S, x−1 A = {y ∈ ϕ(x) : x · y ∈ A}.
(d) A partial semigroup (S, ·) is adequate if and only if for each F ∈ Pf (S),
T
ϕ(x) 6= ∅.
x∈F
T
(e) Given an adequate partial semigroup (S, ·), δS = x∈S c`βS ϕ(x).
5.5 Lemma. Let (S, ·) be an adequate partial semigroup and for p, q ∈ δS define p · q =
A ⊆ S : {x ∈ S : x−1 A ∈ q} ∈ p . Then, with the relative topology inherited from βS,
(δS, ·) is a compact right topological semigroup.
Proof. [2, Proposition 2.6].
onto
5.6 Lemma. Let (S, ·) and (T, ∗) be adequate partial semigroups and let f : S −→T
have the property that for all x ∈ S and all y ∈ ϕS (x), f (y) ∈ ϕT f (x) and f (x · y) =
f (x) ∗ f (y). Let fe : βS → βT be the continuous extension of f . Then the restriction of
fe to δS is a homomorphism from (δS, ·) to (δT, ∗).
Proof. [2, Proposition 2.8].
5.7 Definition. Φ = {f : N → N : f (n) ≤ n for all n ∈ N}.
5.8 Theorem. Let k ∈ N. For each i ∈ {1, 2, . . . , k}, let Ei be a countable commutative
semigroup with identity ei . For each i ∈ {1, 2, . . . , k} and j ∈ N, let hzi,j,t i∞
t=1 be a
sequence in Ei . We assume that, for every i ∈ {1, 2, . . . , k}, zi,1,t = ei for every t ∈ N,
and that hzi,2,t i∞
t=1 is a sequence which contains every element of E i infinitely often.
26
Let ψ be an arbitrary function mapping E1 × E2 × · · · × Ek to a set X and let Ci be
a central set in Ei for each i ∈ {1, 2, . . . , k}. Then, for any finite coloring of X, there
∞
exist a sequence hHn i∞
n=1 in Pf (N), a sequence hci,n in=1 in Ei for each i ∈ {1, 2, . . . , k}
and a monochrome subset A of X such that the following statements hold for every
G ∈ Pf (N), every i ∈ {1, 2, . . . , k} and all f1 , f2 , . . . , fk ∈ Φ:
Q
Q
Q
Q
(i) ψ
n∈G c1,n ·
t∈Hn z1,f1 (n),t , . . . ,
n∈G ck,n ·
t∈Hn zk,fk (n),t ∈ A and
Q
Q
(ii) n∈G ci,n · t∈Hn zi,fi (n),t ∈ Ci .
Proof. Let L = Nk and let v be a “variable” not in L. A located word over L is a function
w from a nonempty finite subset Dw of N to L. Let S0 be the set of located words over
L and let S1 be the set of located variable words over L, that is the set of words over
L ∪ {v} in which v occurs. Let S = S0 ∪ S1 . Given u, w ∈ S, if max Du < min Dw , then
define u · w by Du·w = Du ∪ Dw and for t ∈ Du·w ,
u(t) if t ∈ Du
(u · w)(t) =
w(t) if t ∈ Dw .
With this operation S, S0 , and S1 are adequate partial semigroups so by Lemma 5.5
δS, δS0 , and δS1 , are compact right topological semigroups. Also δS1 is an ideal of
δS. (The verification of this latter statement is an easy exercise and a good chance
for the reader to see whether she has grasped the definition of the operation.) Notice
that for j ∈ {1, 2} and p ∈ βSj , one has that p ∈ δSj if and only if for each n ∈ N,
{w ∈ Sj : min Dw > n} ∈ p.
For each a ∈ L, define θa : S → S0 as follows. For w ∈ S, let Dθa (w) = Dw and for
t ∈ Dw , let
θa (w)(t) =
w(t) if w(t) ∈ L
a
if w(t) = v .
That is, θa (w) is the result of replacing each occurrence of v in w by a. Denote also by
θa its continuous extension taking βS to βS0 and notice that θa is the identity on S0
hence also on βS0 .
Q
For each i ∈ {1, 2, . . . , k}, define gi : S0 → Ei by gi (w) = t∈Dw zi,πi (w(t)),t for
each w ∈ S0 . We shall also use gi to denote the continuous function from βS0 to βEi
which extends gi .
We claim that, if bi ∈ Ei for each i ∈ {1, 2, . . . , k} and if n ∈ N, there exists w ∈ S0
such that gi (w) = bi for every i ∈ {1, 2, . . . , k} and min(Dw ) > n. To see this, observe
that we can choose n1 , n2 , . . . , nk in N such that n < n1 < n2 < . . . < nk and zi,2,ni = bi
for every i ∈ {1, 2, . . . , k}. We can then define w by putting Dw = {n1 , n2 , . . . , nk } and
27
w(ni ) = (1, 1, . . . , 1, 2, 1, . . . , 1), with 2 occurring as the ith term in this k-tuple, for each
i ∈ {1, 2, . . . , k}.
In particular each gi : S0 → Ei is surjective and so, by Lemma 5.6, The restriction
of gi to δS0 is a homomorphism to δEi = βEi .
For each i ∈ {1, 2, . . . , k}, let pi be a minimal idempotent in βEi for which Ci ∈ pi .
We shall first show that we can choose a minimal idempotent q ∈ δS0 and an idempotent
r ∈ δS1 such that q ≤ r, gi (q) = pi for every i ∈ {1, 2, . . . , k}, and θa (r) = q for every
a ∈ L.
Given (X1 , X2 , . . . , Xk , n) ∈ p1 × p2 × · · ·× pk × N we choose w(X1 , X2 , . . . , Xk , n) ∈
S0 such that min(Dw(X1 ,X2 ,...,Xk ,n) ) > n and gi w(X1 , X2 , . . . , Xk , n) ∈ Xi for each
i ∈ {1, 2, . . . , k}. We give p1 × p2 × · · · × pk × N a directed set ordering by stating
that (X1 , X2 , . . . , Xk , n) ≺ (X10 , X20 , . . . , Xk0 , n0 ) if and only if Xi0 ⊆ Xi for each i ∈
{1, 2, . . . , k} and n < n0 . If x is any limit point of the net hw(X1 , X2 , . . . , Xk , n)i in βS0 ,
we have x ∈ δS0 and gi (x) = pi for every i ∈ {1, 2, . . . , k}. (That x ∈ δS0 follows from
the fact that min(Dw(X1 ,X2 ,...,Xk ,n) ) > n. To see that gi (x) = pi , let A ∈ pi and suppose
gi (x) ∈
/ A. Pick B ∈ x such that gi [ B ] ∩ A = ∅. Let Xi = A and for j 6= i let Xj = Ej .
Pick (X10 , X20 , . . . , Xk0 , n0 ) (X1 , X2 , . . . , Xk , 1) such that w(X10 , X20 , . . . , Xk0 , n0 ) ∈ B.
But gi w(X10 , X20 , . . . , Xk0 , n0 ) ∈ Xi0 ⊆ Xi = A, a contradiction.)
Let C = x ∈ δS0 : gi (x) = pi for all i ∈ {1, 2, . . . , k} . We have just seen that C is
nonempty, and so it is a compact subsemigroup of δS0 . Let q be a minimal idempotent
in C. Then q is minimal in δS0 , because if q 0 is any idempotent of δS0 satisfying q 0 ≤ q,
we have gi (q 0 ) ≤ gi (q) = pi for every i ∈ {1, 2, . . . , k}. This implies that gi (q 0 ) = pi for
every i ∈ {1, 2, . . . , k}. So q 0 ∈ C and thus q 0 = q.
Let r be any idempotent in the intersection of the right ideal qδS1 and the left
ideal δS1 q of δS1 . Then r ≤ q. For any a ∈ L, we have θa (r) ≤ θa (q) = q and hence
θa (r) = q.
We define γ : S0 → X by γ (w) = ψ g1 (w), g2 (w), . . . , gk (w) . We can choose a
Tk
monochrome subset A of X such that γ −1 [A] ∈ q. Let Q = γ −1 [A] ∩ i=1 gi −1 [Ci ].
Then Q ∈ q. Let Q? = {w ∈ Q : w −1 Q ∈ q}. Then Q? ∈ q and w −1 Q? ∈ q for every
w ∈ Q? by [13, Lemma 4.14].
We shall inductively choose a sequence hwn i∞
n=1 in S1 such that for each n ∈ N,
(a) if n > 1, then min Dwn > max Dwn−1 and
(b) for every nonempty F ⊆ {1, 2, . . . , n} and every choice of at ∈ {1, 2, . . . , t}k for
Q
t ∈ F , t∈F θat (wt ) ∈ Q? .
28
We first choose w1 ∈ S1 such that θa (w1 ) ∈ Q? , where a denotes the k-tuple
(1, 1, . . . , 1). This is possible because θa −1 [Q? ] ∈ r and so θa −1 [Q? ] 6= ∅. Now let n ∈ N
and assume that w1 , w2 , . . . , wn have been chosen. Let
Q
k
U=
(w
)
:
∅
=
6
F
⊆
{1,
2,
.
.
.
,
n}
and
for
all
t
∈
F
,
a
∈
{1,
2,
.
.
.
,
t}
.
θ
a
t
t
t
t∈F
T
By our assumption (b), U ⊆ Q? so u∈U u−1 Q? ∈ q. We observe that, for any V ∈ q
and any a ∈ L, θa −1 [V ] ∈ r and that {w ∈ S1 : min(Dw ) > max(Dwn )} ∈ r. Thus we
can choose wn+1 such that min(Dwn+1 ) > max(Dwn ), and
T −1 ? T
wn+1 ∈
θa
Q ∩ u∈U u−1 Q? : a ∈ {1, 2, . . . , n + 1}k .
We can now conclude the proof. For each n ∈ N and i ∈ {1, 2, . . . , k}, let Hn =
Q
{t ∈ Dwn : wn (t) = v} and let ci,n = t∈Dw \Hn zi,πi (wn (t)),t . Then, if a ∈ L, we have
n
Q
gi θa (wn ) = ci,n · t∈Hn zi,πi (a),t .
Suppose now that f1 , f2 , . . . , fk ∈ Φ and G ∈ Pf (N). For each n ∈ N, define
an ∈ {1, 2, . . . , n}k by πi (an ) = fi (n) for each i ∈ {1, 2, . . . , k}. Then for each i ∈ {1, 2,
. . . , k}, we have
Q
Q
Q
Q
n∈G ci,n ·
t∈Hn zi,fi (n),t =
n∈G ci,n ·
t∈Hn zi,fi (n),t
Q
= n∈G gi θan (wn )
Q
= gi
n∈G θan (wn ) .
Q
Since n∈G θan (wn ) ∈ Q, γ [Q] ⊆ A, and gi [Q] ⊆ Ci for each i ∈ {1, 2, . . . , k} the
conclusions of the theorem hold.
We show now how to derive a simple strengthening of Theorem 4.8 from Theorem
5.8.
5.9 Corollary. Let m, k ∈ N. Let C1 be central in (N, +) and let C2 be central in (N, ·).
Sm
∞
For each i ∈ {0, 1, . . . , k} let hxi,t i∞
t=1 and hyi,t it=1 be sequences in N. Let N =
s=1 As .
Then there exist s ∈ {1, 2, . . . , m}, F ∈ Pf (N), and a, b ∈ N such that
P
{ba}
∪
b(a
+
x
)
:
i
∈
{0,
1,
.
.
.
,
k}
∪
i,t
t∈F
Q
1, . . . , k} ∪
t∈F yj,t : j ∈ {0,
ba · P
Q
b(a + t∈F xi,t ) · ( t∈F yj,t ) : i, j ∈ {0,
1,
.
.
.
,
k}
⊆ As ,
P
{a} ∪ a + t∈F xi,t : i ∈ {0, 1, . . . , k}
⊆
C
,
and
1
Q
{b} ∪ b · t∈F yj,t : j ∈ {0, 1, . . . , k} ⊆ C2 .
Proof. Let E1 = (ω, +) and let E2 = (N, ·). Define ψ : E1 × E2 → ω by ψ(a, b) = ab.
For t ∈ N let z1,1,t = 0 and z2,1,t = 1. For i ∈ {1, 2} let hzi,2,t i∞
t=1 be a sequence
which contains every element of Ei infinitely often. For j ∈ {0, 1, . . . , k} and t ∈ N let
29
z1,j+3,t = xj,t and z2,j+3,t = yj,t . (For j > k + 3 we do not care what z1,j,t and z2,j,t
are.)
∞
∞
Since N is an ideal of (ω, +), C1 is central in E1 . Pick hHn i∞
n=1 , hc1,n in=1 , hc2,n in=1 ,
and A as guaranteed by Theorem 5.8. Pick s ∈ {1, 2, . . . , m} such that A ⊆ As . Let
n = k + 3. (We choose n = k + 3 rather than n = 1 so that there will be functions f 1
and f2 in Φ with the properties required of them below.) Let a = c1,n , let b = c2,n ,
P
and let F = Hn . If f1 (n) = 1, then c1,n + t∈Hn z1,f1 (n),t = a. If f1 (n) = j + 3
P
P
for some j ∈ {0, 1, . . . , k}, then c1,n + t∈Hn z1,f1 (n),t = a + t∈F xj,t . If f2 (n) = 1,
Q
then c2,n · t∈Hn z2,f2 (n),t = b. If f2 (n) = j + 3 for some j ∈ {0, 1, . . . , k}, then
Q
Q
c2,n · t∈Hn z2,f2 (n),t = b · t∈F yj,t .
We conclude with a simple variation on the proof of Theorem 5.8 which applies in
case the semigroups are all the same.
5.10 Theorem. Let k ∈ N, let E be a countable commutative semigroup with identity
e, let R1 , R2 , . . . , Rk be IP-sets in E, and let C be a central subset of E. There exist
ri ∈ Ri and bi ∈ E for each i ∈ {1, 2, . . . , k} such that whenever f : {1, 2, . . . , k} →
{1, 2, . . . , k}, h : {1, 2, . . . , k} → {0, 1, . . . , k}, and ∅ 6= F ⊆ {1, 2, . . . , k}, one has
Q
h(i)
∈ C.
i∈F bi · (rf (i) )
k
Proof. Let L = {1, 2, . . . , k 2 + k + 2} and let v, S0 , S1 , S, hDw iw∈S , and hθa ia∈L
be as in the proof of Theorem 5.8. For j ∈ {1, 2, . . . , k} pick a sequence hxj,t i∞
t=1 such
∞
that F P (hxj,t it=1 ) ⊆ Rj . For m ∈ {0, 1, . . . , k}, j ∈ {1, 2, . . . , k}, and t ∈ N, let
z2+mk+j,t = (xj,t )m . Let z1,t = e for each t and let hz2,t i∞
t=1 be a sequence in E which
takes on each member of E infinitely often.
Q
For i ∈ {1, 2, . . . , k}, define gi : S0 → E by gi (w) = t∈Dw zπi (w(t)),t . For F ∈
Q
Pf ({1, 2, . . . , k}), define γ F : S0 → E by γ F (w) = i∈F gi (w) (so γ {i} = gi ). Denote
also by γ F the continuous extension taking βS0 to βE.
As in the proof of Theorem 5.8 we see that given any b1 , b2 , . . . , bk ∈ E there
is some w ∈ S0 such that gi (w) = bi for each i ∈ {1, 2, . . . , k}. In particular each
γ F is a surjective homomorphism so by Lemma 5.6 the restriction of γ F to δS0 is a
homomorphism to βE.
Pick a minimal idempotent p ∈ βE such that C ∈ p. We claim that for any
B ∈ p and any n ∈ N there exists wB,n ∈ S0 such that for all F ∈ Pf ({1, 2, . . . , k}),
γ F (wB,n ) ∈ B. To see this pick b1 , b2 , . . . , bk such that F P (hbt ikt=1 ) ⊆ B, which one may
do because p is an idempotent. Pick wB,n such that gi (wB,n ) = bi for each i ∈ {1, 2,
. . . , k}.
30
Direct D = {(B, n) : B ∈ p and n ∈ N} by (B, n) ≺ (B 0 , n0 ) if and only if B 0 ⊆ B
and n < n0 . Let u be a limit point of the net hwB,n i(B,n)∈D in βS0 . We see as in the
proof of Theorem 5.8 that u ∈ δS0 and γ F (u) = p for all F ∈ Pf ({1, 2, . . . , k}). Let J =
{w ∈ δS0 : γ F (w) = p for all F ∈ Pf ({1, 2, . . . , k})}. Then J is a compact subsemigroup
of δS0 since each γ F is a continuous homomorphism. Pick a minimal idempotent q of
J . Given any idempotent q 0 ∈ δS0 such that q 0 ≤ q, for each F ∈ Pf ({1, 2, . . . , k}),
γ F (q 0 ) ≤ γ F (q) = p so γ F (q 0 ) = p. Thus q 0 ∈ J and so q 0 = q. That is, q is minimal in
δS0 .
Now we claim that we may choose w ∈ S1 such that γ F θa (w) ∈ C for every
a ∈ L and every F ∈ Pf ({1, 2, . . . , k}). To see this, pick an idempotent r in qδS1 ∩ δS1 q.
Then r ≤ q so for each a ∈ L, θa (r) ≤ θa (q) = q and so θa (r) = q and thus for each
T
F ∈ Pf ({1, 2, . . . , k}), γ F θa (r) = γ F (q) = p. Pick w ∈ S1 ∩ {(γ F ◦ θa )−1 [C] : a ∈ L
and F ∈ Pf ({1, 2, . . . , k})}.
Q
Let H = {t ∈ Dw : w(t) = v}. For i ∈ {1, 2, . . . , k}, let bi = t∈Dw \H zπi (w(t)),t
Q
and let ri = t∈H xi,t . Now let f : {1, 2, . . . , k} → {1, 2, . . . , k}, h : {1, 2, . . . , k} →
{0, 1, . . . , k}, and ∅ 6= F ⊆ {1, 2, . . . , k}. Let
a = 2 + h(1)k + f (1), 2 + h(2)k + f (2), . . . , 2 + h(k)k + f (k) .
Then for i ∈ F ,
so
Q
i∈F
bi (rf (i) )h(i)
Q
bi (rf (i) )h(i) = bi · ( t∈H (xf (i),t )h(i)
Q
= bi · t∈H zπi (a),t
= gi θa (w)
= γ F θa (w) ∈ C.
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32
Mathias Beiglböck
Institute of Discrete Mathematics and Geometry
Vienna University of Technology
Wiedner Hauptstr. 8-10
1040 Wien
Austria
[email protected]
Vitaly Bergelson
Department of Mathematics
Ohio State University
Columbus, OH 43210
USA
[email protected]
Neil Hindman
Department of Mathematics
Howard University
Washington, DC 20059
USA
[email protected]
Dona Strauss
Department of Pure Mathematics
University of Hull
Hull HU6 7RX
UK
[email protected]
33